D-modules et cycles évanescents (d’après B. Malgrange et M. Kashiwara). (D-modules and vanishing cycles (following B. Malgrange and M. Kashiwara)). (French) Zbl 0623.32013

Sabbah, C.

Géométrie algébrique et applications, C. R. 2ième Conf. int., La Rabida/Espagne 1984, III: Géométrie réelle. Systèmes différentiels et théorie de Hodge, Trav. Cours 24, 53-98 (1987).

[For the entire collection see Zbl 0614.00007.]
This paper is an expository paper of the work by B. Malgrange and M. Kashiwara on regular holonomic \({\mathcal D}_ X\)-modules. Let X be a complex manifold, \({\mathcal D}_ X\) the sheaf of differential operators on X with holomorphic coefficients and f:X\(\to {\mathbb{C}}^ a \)holomorphic map. B. Malgrange constructed a regular holonomic \({\mathcal D}_ X\)-module corresponding to f and prove that its de-Rham complex (through Riemann- Hilbert correspondence) is quasi-isomorphic to a complex of cycles évanescents of constant sheaf. Kashiwara proved that such property is true for all regular holonomic \({\mathcal D}_ X\)-modules. The author of this paper begins with the definition of good filtrations and gives a brief explanation of cycles évanescents and its relation with \({\mathcal D}_ X\)-modules. In the last section the author gives an application of the result for calculation of cycles évanescents of the constant sheaf when f is a function with normal crossing.

Reviewer: M.Muro

MSC:

32C38	Sheaves of differential operators and their modules, \(D\)-modules
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Keywords:

Hodge theory; vanishying cycles; regular holonomic \({\mathcal D}_ X\)-module

Citations:

Zbl 0614.00007